Econometric Institute Report No. 9720A ON PURCHASE

EconometricInstituteReportNo.9720/AONPURCHASETIMINGMODELSINMARKETINGJ.B.G.FrenkandS.ZhangEconometricInstituteErasmusUniversityRotterdamMay,1997ABSTRACTInthispaperweconsiderstochasticpurchasetimingmodelsusedinmarketingforlow-involvementproductsandshowthatimportantcharacteristicsofthosemodelsareeasytocompute.Assuchthesecalculationsarebasedonanelementaryprobabilisticargumentandcovernotonlythewell-knowncondensednegativebinomialmodelbutalsostochasticpurchasetimingmodelswithotherinterarrivalandmixingdistributions.Keywords:Marketing,purchasetimingmodel.AMSsubjectclassication:90A60,60G07.1IntroductionInthispaperweconsiderpurchasetimingmodelsusedwithinthemarketingliterature(cf.[8])andshowbyeasyargumentshowtocomputesomeimportantcharacteristicsofthesemodelsundervariousassumptionsonthemixingdistributionandtheassociated\standard-izedpurchasetimingprocess.AfterintroducingageneralframeworkforthesemodelswediscussapurchasetimingmodelwithanErlang-rmixingdistributionandanarbitrarypointprocessrepresentingthis\standardizedpurchasetimingprocess.Alsoweconsiderapur-chasetimingmodelwithanarbitrarymixingdistributionandanErlang-srenewalprocessasa\standardizedpurchasetimingprocess.Forthelastclassofmodelsitisrelativelyeasytoderiveanalyticalformulasfortheimportantcharacteristicsandtheseformulasgen-eralizemostoftheresultsavailableintheliterature.Atthesametimeweshowthatthemathematicsinvolvedisquiteelementary.2PurchasetimingmodelsLetfXi:i1gdenoteasequenceofnonnegativerandomvariablesandconsidertheassociatednonexplosiveunivariatepointprocessfN(t):t0ggivenbyN(t):=supfn0:TntgwithTn,n2N,denotingthesumoftherandomvariablesXi,1in,andT0:=0(cf.[2]).ObserveiftherandomvariablesXi,i=1;:::,areindependentandidenticallydistributedwithdistributionfunctionF(x):=PrfXixgsatisfyingF(0)=0theabovepointprocessrepresentsarenewalprocess(cf.[11]).TomodelthemomentsofpurchasetimingofacustomerselectedatrandomfromapopulationitisassumedthattheinterpurchasetimesofthisrandomcustomeraregivenbyXi=Y,i2N,withYanonnegativerandomvariablewithdistributionG(y):=PrfYyg.Thisdistributioniscontinuouson(0;1)andsatises0G(0)1andG(1)=1.Moreover,therandomvariableYrepresentingthepurchaserateparameter(cf.[8])isindependentofthesequenceXi,i1.WithinthetheoryofconsumerbehaviorthedistributionGiscalledthemixingdistributionandthisdistributionenablesustoaggregateoverthewholepopulationofcustomers.ObservealsothatinmostoftheliteratureonconsumerbehaviortheunivariatepointprocessfN(t):t0gisactuallyarenewalprocesswitheitheranexponentialorErlang-2interarrivaldistribution.IntroducingnowthestochasticprocessfBt:t0ggivenbyBt:=thenumberofpurchasesofarandomcustomeruptotimet1itfollowsbytheaboveconstructionthatBt=N(Yt).Awell-knownmodelbelongingtothisclassisgivenbytheNegativeBinomialmodel(NBD)(cf.[8,5,9]).InthismodelitisassumedthatthemixingdistributionisaGammadistributionandtheassociatedpointprocessisaPoissonprocesswitharrivalrate1.FromatheoreticalpointofviewimportantcharacteristicsoftherandomvariableBtareitsdistribution,itsrstmomentandgeneratingfunction.TocomputethedistributionofBtweobserve,sincetheeventfN(t)kg,k2N,coincideswiththeeventfTktg,thatPrfBtkg=PrfN(Yt)kg=PrfTkYtg=PrfYTkt1g:SinceGiscontinuouson(0;1)andTkisstrictlypositivewithprobabilityoneweobtainthatPrfBtkg=PrfYTkt1g=1EG(Tkt1)(2.1)withEdenotingtheexpectation.IfithappensthattheconsideredpopulationconsistsofmdierentclasseseachcharacterizedbyadierentrandompurchaserateparameterYi;i=1;:::;m,themixingdistributionGcanbeseenasamixtureofdistributions.Thismeansthatthereexistpositivenumbersp1;:::;pmaddingupto1withpirepresentingtherelativesizeofclassiwithinthepopulationandeachrandomcustomerbelongingtoclassihasarandompurchaserateparameterYiwithdistributionGi.HenceinthiscasethemixingdistributionGisgivenbyG(y)=mXi=1piGi(y)orequivalentlyGisthedistributionoftherandomvariableYIwhereIdenotesarandomvariablewithPrfI=ig=pi,i=1;:::;mandIisindependentoftherandomvariablesY1;:::;Ym.By(2.1)wenowobtainthatPrfBtkg=PrfN(YIt)kg=mXi=1piPrfN(Yit)kg=mXi=1piPrfB(i)tkg(2.2)withB(i)tdenotingthenumberofpurchasesuptotimetofacustomerselectedatrandomfromclassi.Aspecialcaseisgivenbytheexistenceofazeroandanonzero-classwithinthepopulationandby(2.2)thisimpliesthatPrfBtkg=(1p1)PrfB(2)tkg+p10(k)with0(k)=1fork=0andzerootherwiseandB(2)tdenotingthenumberofpurchasesuptotimetofacustomerselectedatrandomfromthenon-zeroclass.Fromatheoreticalpointofviewthereseemstobenopreferenceforaspecicmixingdistributionandso2theselectionofsuchadistributionispurelydeterminedbytheexibilityofthefamilyofdistributionstowhichthismixingdistributionbelongs.SincethefamilyofGammadistributionswithscaleparameter0andshapeparameter0seemstobeexibleenoughtheGammadistributionischoseninmostoftheliterature(forexamplesee[14,3])asamixingdistribution.IftheshapeparameterisanintegerrthecorrespondingGammadistributioniscalledanErlang-rdistributionandinthiscasethecorrespondingrandomvariableYcanberepresentedasthesumofrindependentandexponentiallydistributedrandomvariablesYi,i=1;:::;r,withthesamescaleparamet